I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked".

So the question: What are good books, for laymen, which teach interesting Mathematics, but actually does it in a "real" way. For example, "Fermat's Last Enigma" doesn't count, since it doesn't actually feature any Maths, just a story, and most textbook don't count, since they don't feature a story.

My favorite example of this is "Journey Through Genius", which is a brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level.

Edit:

A few more details on what I'm looking for.

The audience of "laymen" should be anyone who has the ability (and desire) to understand actual mathematics, but does not want to learn from a textbook. Obviously I'm thinking about myself here, as a programmer who loves mathematics, I love being exposed to real maths, but I'm not going to get into it seriously. That's why books that show actual maths, but give a lot more exposition (and much clearer explanations, especially of what the intuition should be) are great.

When I say "real maths", I'm talking about actual proofs, formulas, or other mathematical theories. Specifically, I'm not talking about philosophy, nor am I talking about books which only talk about the history of maths (Simon Singh style), since they only talk about maths, they don't actually show anything. William Dunham's books and Paul J. Nahin's books are good examples.

I think there will be good answers to this, but it might be helpful to give a few more specific conditions on who the layman is (five year old with a parent's help? computer engineer? etc) and what counts as maths (does a discussion of Escher's work, for example, count, or people like Russell or Boole on philosophy of math?) It's hard for someone to rank which responses they think work best for such a broad question.
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Jamie BanksJul 21 '10 at 7:17

It's been on my list of must-read books for a long time. Unfortunately, I don't have anywhere near the time to read it right now, and it doesn't cost pocket change either.
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Edan MaorJul 30 '10 at 16:45

It took me about one and a half year on and off to read it. One problem is that it's too heavy to carry when traveling.
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starblueJul 30 '10 at 20:49

I've been successfully using Courant and Robbins, What is Mathematics, on adults who have not had a math class for a few decades, but are open to the idea of learning more about mathematics.

Some sections are too advanced for someone with only high school mathematics, and many more will appear that way to the person at first, but do not actually rely on anything beyond high school mathematics.

I think that a non-mathematician could appreciate T.W.Körner's book The Pleasures of Counting; but I still believe that the collection of "Mathematical Games" columns from Martin Gardner are the very best thing.

One danger worth mentioning is that Nahin's books are full of mistakes and inaccuracies. He is known for "lying" about certain things that he perceives as too difficult to explain, rather than alerting the reader. In addition to this, he is absolutely uninterested in fixing these errors in later editions of his books, even when compact solutions are noted.
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BBischofJul 21 '10 at 22:45

Yes, the later chapters of Dr. Euler's Fabulous Formula (on Fourier transforms, a subject I know a good amount about) were incomprehensible to me. Still, the first half or so of the book is very good, and full of wonderful gems.
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BlueRaja - Danny PflughoeftJul 22 '10 at 5:42

Written and illustrated(Pictures are great ;p) by a couple: Lillian R. Lieber, and Hugh Gray Lieber. These books were hard to find before because they went out of print but I have this new version and like it a lot. The books explains profound topics in a way that is graspable by anyone without being dumbed down.

Godel's proof is one I enjoyed. It's was a little hard to understand but there is nothing in this book that makes it inaccessible to someone without a strong math background.

John Derbyshire's Prime Obsession is about Riemann's hypothesis. One of the stated goals of the author is to explain what "all non-trivial zeros of the zeta function have real part one-half" means to readers who have no background in calculus. Odd-numbered chapters tell the story of how Riemann came to his hypothesis, and even-numbered chapters are more mathematical in nature.

I think Mathematics : A very short Introduction by Timothy Gowers is a very interesting read. As such, it's not something you "learn" from, but it makes for a very short and sweet introduction to someone who is just curious about mathematics. (Also its also a great read for people who actually do work in mathematics, because that's where I lift my examples from, when I explain to my family and friends what I am doing!) ;)

I also recommend James R. Newman's The world of mathematics. But be warned, while the content is not very technical (indeed, many articles in the collection are from public lectures of famous scientists), it can get a little bit dry at times. If you are patient enough for it, it is a very good companion to Courant and Robbins' What is mathematics. (It is sort of like the Princeton Companion, but older and slightly more down-to-earth).

Lastly, you can also try the various books and articles by Brian Hayes.

I'm surprised nobody has mentioned Mathematics and the Imagination by Edward Kasner and James Newman; originally published in 1940, still available in an edition from Dover. Among other things, the book that introduced the world to the name "googol" for $10^{100}$. It's a classic, and I've never heard anything bad about it. The book is meant for non-mathematicians.

I second the recommendation of Martin Gardner's columns as a follow-up.

A more recent addition to this genre is The Calculus Diaries: How math can help you lose weight, win in Vegas, and survive zombie apocalypse by Jennifer Ouellette; it was reviewed favorably in NPR's "Science Friday"; written by a non-mathematician who never got through Calculus in school, also for non-mathematicians. I've heard some minor criticisms of the style, but otherwise generally positive reviews.

Definitely, "Elementary Differential Equations" by W.Boyce and R. Di Prima, as it covers a mathematical subject which is used in many applied science disciplines in a way that makes it understandable to non-mathematicians.

As an undergrad, I read a fair number of pop math books. The best by far was Ash and Gross' "Fearless Symmetry". This book is very beautiful. It sustains a nice level of rigor while being approachable by those who aren't professionals. Additionally, it weaves the tale of one of the most beautiful recent stories in mathematics. Everyone I know of who have read the book have found it wonderful.

There are good reasons not to recommend this book, especially to a non-mathematician who is as yet unaware of alternative points of view. Some reasons why are listed here. I personally think those critiques are weak compared to what could be said.
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Dan PiponiJul 21 '10 at 18:09

In it Mlodinow (a physicist) explains randomness as well as statistical principles and methods; he is very good at pointing out misconceptions about randomness and providing clear examples of how and why you should avoid them.

I've read and enjoyed some of the titles mentioned by Dunham (who balances motivations and mathematics really well; Euler: The Master of Us All was particularly good) and Stewart. Maor and Nahin also have some decent accounts. The former's book Trigonometric Delights sparked an interest in me on the history of maths back when I read it during high school, but it doesn't shy from actual derivations and mathematical reasoning.

Possibly the best book I've come across of this type however is Julian Havil's Gamma: Exploring Euler's Constant. It was one of the first such accounts I'd read, and revisiting it recently I found it just as informative as ever. In place of biography alone (though there are plenty of fascinating historical and anecdotal titbits), Havil investigates connections through mathematics via the more mysterious of 'Euler's constants' (the Euler-Mascheroni constant as it's called). It's somewhat like Nahin's book on $i$, but Havil's treatment is more cautious and farther-reaching, if slightly more demanding.

I rather enjoyed Professor Stewart's book [1]. Take a look at it; I hope you enjoy it.

I have blogged about a selection from his book, you can view it at [2]. This is just one of the many different mathematical concepts covered in the book. It is more of a fun book than lots of theory. It will get you to think.

Same as me. After so far and so many the books I've read about math. I think the best option for is "Mathematics - Its Content, Methods, and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev. It's a russian translated book. It came with 3 part. The book cover all major topic in maths. That is the best maths book I ever found. :-). And here some advice from another maths enthuast https://medium.com/@amathstudent/learning-math-on-your-own-39fe90c3536b.

My recommendation is Great Formulas Explained by Metin Bektas (as well as the second volume More Great Formulas Explained). It's a nice collection of formulas from mathematics, physics and economics for non-mathematicians with lots of examples on how and where to apply them. A little bit of algebra is sufficient to follow the text.